Scaling Limits of Jacobi Matrices and the Christoffel–Darboux Kernel

Abstract

We study scaling limits of deterministic Jacobi matrices, centered around a fixed point $x_0$, and their connection to the scaling limits of the Christoffel–Darboux kernel at that point. We show that in the case when the orthogonal polynomials are bounded at $x_0$, a subsequential limit always exists and can be expressed as a canonical system. We further show that under weak conditions on the associated measure, bulk universality of the CD kernel is equivalent to the existence of a limit of a particular explicit form.

6 Appendix: Canonical Systems, Jacobi Matrices, and the CD Kernel

Canonical systems generalize many second order difference and differential operators, with Schrödinger, Dirac, and Jacobi being particular cases. Furthermore, the correspondence between such systems and Hermite–Biehler functions (see below for a definition), established by de Branges [6], is a central result in the theory of de Branges spaces. Thus there is a huge literature on canonical systems, spanning spectral theory, harmonic analysis and number theory ([1, 6, 8, 13, 18, 27, 28] are only a few relevant references). The next several paragraphs contain only a quick review of some results that are directly relevant here (for details, see, e.g., [27]).

A canonical system is a family (indexed by $z \in \mathbb {C}$) of differential equations of the form

$$\begin{aligned} \mathcal {J}u'(t)=z H(t)u(t) \end{aligned}$$

(6.1)

on some interval $I=[0,L] \subseteq \mathbb {R}$, where $\mathcal {J}=\left( \begin{array}{cc} 0 &{} -1 \\ 1 &{} 0 \end{array} \right)$ and H(t) is a $2\times 2$ nonnegative definite matrix valued function such that the entries of H are integrable functions on I. By a change of variable (see, e.g., [27, Section 6] or [28]), we may also assume that $H(t) \not \equiv 0$ on nonempty open sets.

In the case that H(t) is invertible almost everywhere, we may rewrite (6.1) as

$$\begin{aligned} H^{-1}(t)\mathcal {J} u'(t)=z u(t), \end{aligned}$$

i.e., as an eigenvalue equation for the operator $H^{-1}(t)\mathcal {J}\frac{d}{dt}$ which is symmetric with respect to the inner product

$$\begin{aligned} \left( f,g \right) _H=\int _0^L \left( f(t),H(t) g(t)\right) _{\mathbb {C}^2}\text {d}t. \end{aligned}$$

(6.2)

Let $\mathcal {H}_H$ be the Hilbert space of vector valued functions on I corresponding to this inner product. Choosing appropriate boundary conditions at 0 and L (e.g., $f(0)=f(L)=\left( \begin{array}{c} 1 \\ 0 \end{array} \right)$) defines a domain of self-adjointness for this operator. Moreover, even if H is not invertible a.e., as long as the boundary condition at 0 is not orthogonal to $\text {Image}(H)(t)$ $\forall t\in (0,\varepsilon )$ for some $\varepsilon >0$ (and a similar condition is satisfied at L), it is possible to define a subspace of $\mathcal {H}_H$ such that (6.1) is the eigenvalue equation for a self-adjoint operator defined on that subspace (see [28, Section 2]).

A solution to (6.1) is an absolutely continuous function, $u: [0,L]\rightarrow \mathbb {C}^2$, that satisfies (6.1) a.e.

Let u(t, z) be a solution with initial value $u(0)=\left( \begin{array}{c} 1 \\ 0 \end{array} \right)$. Then the function $E_L(z)=u_1(L,z)+iu_2(L,z)$ is a Hermite–Biehler function; i.e., it has no zeros in the upper half plane $\mathbb {C}^+=\left\{ z \mid \text {Im}z>0 \right\}$ and satisfies

$$\begin{aligned} \left| E_L(z) \right| \ge \left| E_L(\overline{z}) \right| \text { for } z \in \mathbb {C}^+. \end{aligned}$$

Such functions are at the basis of the theory of de Branges spaces.

The de Branges space B(E)associated with a Hermite–Biehler function E is the set of all entire functions, f, such that both $\frac{f}{E}$ and $\frac{f^\sharp }{E}$ are in $H^2(\mathbb {C}^+)$, where $f^\sharp (z)=\overline{f(\overline{z})}$. It is a reproducing kernel Hilbert space with inner product given by

$$\begin{aligned} \left( f,g \right) _E=\frac{1}{\pi }\int _\mathbb {R} \overline{f(x)}g(x)\frac{\text {d}x}{|E(x)|^2} \end{aligned}$$

and reproducing kernel

$$\begin{aligned} K_E(z,\zeta )=\frac{\overline{E(z)}E(\zeta )-E(\overline{z})\overline{E(\overline{\zeta })}}{2i(\overline{z}-\zeta )}. \end{aligned}$$

It follows that every canonical system (6.1) gives rise to a de Branges space through the function $E_L(z)$ associated with its solution as described above. In fact, in this particular case, it is not hard to show that the reproducing kernel is also given by

$$\begin{aligned} K_{E_L}(z,\zeta )=\frac{1}{\overline{z}-\zeta }\det \left( \begin{array}{cc} u_1(L,\overline{z}) &{} u_1(L,\zeta ) \\ u_2(L,\overline{z}) &{} u_2(L,\zeta ) \end{array} \right) \end{aligned}$$

(6.3)

and by

$$\begin{aligned} K_{E_L}(z,\zeta )=\left( u(\cdot ,z),u(\cdot ,\zeta ) \right) _H=\int _0^L \left( u(t,z),H(t) u(t,\zeta )\right) _{\mathbb {C}^2}\text {d}t. \end{aligned}$$

(6.4)

In particular, for any fixed z, $K_{E_L}(z,\zeta )$ is an entire function of $\zeta$.

It turns out that this is not a particular example, but rather the general case: a fundamental result in the theory of de Branges spaces (see, e.g. [6, 27, 28]) says that every de Branges space is associated with a canonical system.

As shown in [24, Section 4], the CD kernel $K_n(x,y)$ associated with a measure $\mu$ is a reproducing kernel for a de Branges space as well. Indeed, let

$$\begin{aligned} L_n(x,y)=(x-y)K_n(x,y). \end{aligned}$$

Then the Christoffel–Darboux formula (2.9) says that

$$\begin{aligned} L_n(x,y)=a_n\left( p_n(x)p_{n-1}(y)-p_n(y)p_{n-1}(x) \right) . \end{aligned}$$

(6.5)

Now for any fixed $w \in \mathbb {C}^+$, the function

$$\begin{aligned} E_{n,w}(\cdot )=\sqrt{2}\frac{L_n(\overline{w},\cdot )}{\left| L_n(w,\overline{w}) \right| ^{1/2}} \end{aligned}$$

is a Hermite–Biehler function. The corresponding de Branges space, $B(E_{n,w})$, is the space of polynomials of degree $<n$ and its reproducing kernel is

$$\begin{aligned} K_{E_{n,w}}(z,\zeta )=K_n(\overline{z},\zeta ) \end{aligned}$$

(as can be seen easily from (2.9)). Note that the definition in [24, Theorem 4.3] differs from ours by a factor of $\sqrt{\pi }$; this is because we have an extra factor of $\pi$ in the inner product defining B(E). As shown in Sect. 2 above, this de Branges space is naturally associated with the discrete canonical system (2.7) with $x_0=0$.

It is an interesting fact that it is possible to also go in the other direction and associate a Jacobi matrix with any discrete canonical system satisfying the appropriate conditions. Let $\{r_\ell \}_{\ell =0}^\infty$ and $\{s_\ell \}_{\ell =0}^\infty$ be two real sequences satisfying

$$\begin{aligned} s_\ell r_{\ell -1}-r_{\ell }s_{\ell -1}=\frac{1}{a_\ell } \end{aligned}$$

for some sequence of positive numbers $\{a_\ell \}_{\ell =0}^\infty$ with $a_0=1$. Consider the discrete canonical system

$$\begin{aligned} \begin{aligned}&\mathcal {J}\left( \widehat{u}_{\ell +1}(x)-\widehat{u}_\ell (x)\right) \\&\quad =x\left( \begin{array}{cc} r_\ell ^2 &{} -s_\ell r_\ell \\ -s_\ell r_\ell &{} s_\ell ^2 \end{array} \right) \widehat{u}_\ell (x). \end{aligned} \end{aligned}$$

Then if $\{u_\ell (x)\}$ is a solution satisfying $u_0=\left( \begin{array}{c} 1 \\ 0 \end{array} \right)$, then

$$\begin{aligned} p_\ell (x)=r_\ell u_{\ell ,1}(x)-s_\ell u_{\ell ,2}(x) \end{aligned}$$

is the $\ell$’th orthonormal polynomial with respect to the Jacobi matrix whose off diagonal parameter sequence is the given sequence $\{a_\ell \}_{\ell =1}^\infty$, and whose diagonal entries are

$$\begin{aligned} b_\ell =a_\ell a_{\ell -1}\left( r_\ell s_{\ell -2}-s_\ell r_{\ell -2} \right) . \end{aligned}$$

That this is true follows by a direct computation writing

$$\begin{aligned} \widehat{T}_{\ell }(0)=\left( \begin{array}{cc} a_\ell r_\ell &{} -a_\ell s_\ell \\ r_{\ell -1} &{} -s_{\ell -1} \end{array} \right) \end{aligned}$$

and noting that $\widehat{T}_\ell (0)\widehat{T}_{\ell -1}^{-1}(0)=S_\ell (0)$, from which one may compute the values of the Jacobi parameters. For a similar analysis in the continuum, associating canonical systems to one-dimensional Schrödinger operators, see [27, Section 8].